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'''Bohlen-Pierce-Stearns''' ( | '''Bohlen-Pierce-Stearns''' (BPS) is a [[temperament]] in the 3.5.7 [[subgroup]] generated by a sharp [[~]][[9/7]] (or equivalently a flat ~[[7/3]]), [[tempering out]] the sensamagic comma, [[245/243]] so that a stack of two generators represents [[5/3]] in addition to 81/49, which generates a [[MOS scale]] of [[4L 5s (3/1-equivalent)|4L 5s]] against the tritave, known as the Bohlen-Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13]]edt, representing the equal-tempered [[Bohlen-Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17]]edt, [[30edt|7\30]]edt, and [[43edt|10\43]]edt. | ||
For technical data, see [[Sensamagic clan#BPS]]. | As the generator of the Bohlen-Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad. | ||
For technical data, see ''[[Sensamagic clan#BPS]]''. | |||
== Extensions == | |||
Several extensions of this temperament are possible to incorporate additional harmonics. | |||
== Interval chains == | |||
<div><div style="display: inline-grid; margin-right: 25px;"> | |||
{| class="wikitable center-1 right-2" | |||
|+ style="font-size: 105%;" | Basic 3.5.7 temperament | |||
|- | |||
! # !! Cents* !! Approximate Ratios | |||
|- | |||
| 0 || 0.0 || '''1/1''' | |||
|- | |||
| 1 || 440.6 || 9/7, 35/27 | |||
|- | |||
| 2 || 881.1 || 5/3, 81/49 | |||
|- | |||
| 3 || 1321.7 || 15/7, 175/81 | |||
|- | |||
| 4 || 1762.2 || 25/9, 135/49 | |||
|- | |||
| 5 || 300.8 || 25/21, 405/343, 875/729 | |||
|- | |||
| 6 || 741.4 || 75/49, 125/81 | |||
{{table notes|cols=3 | |||
| In 3.5.7-subgroup [[DKW theory|DKW]] tuning | |||
}} | |||
|} | |||
</div> | |||
=== Other tunings === | |||
* [[DKW theory|DKW]] (3.5.7): ~3 = 1\1, ~9/7 = 440.554 | |||
[[Category:Temperaments]] | [[Category:Temperaments]] | ||
Revision as of 08:46, 22 August 2024
Bohlen-Pierce-Stearns (BPS) is a temperament in the 3.5.7 subgroup generated by a sharp ~9/7 (or equivalently a flat ~7/3), tempering out the sensamagic comma, 245/243 so that a stack of two generators represents 5/3 in addition to 81/49, which generates a MOS scale of 4L 5s against the tritave, known as the Bohlen-Pierce Lambda scale. The "canonical" tuning for the generator is 3\13edt, representing the equal-tempered Bohlen-Pierce scale, but a range of other tunings are valid, including 4\17edt, 7\30edt, and 10\43edt.
As the generator of the Bohlen-Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to meantone of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.
For technical data, see Sensamagic clan#BPS.
Extensions
Several extensions of this temperament are possible to incorporate additional harmonics.
Interval chains
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 440.6 | 9/7, 35/27 |
| 2 | 881.1 | 5/3, 81/49 |
| 3 | 1321.7 | 15/7, 175/81 |
| 4 | 1762.2 | 25/9, 135/49 |
| 5 | 300.8 | 25/21, 405/343, 875/729 |
| 6 | 741.4 | 75/49, 125/81 |
Other tunings
- DKW (3.5.7): ~3 = 1\1, ~9/7 = 440.554
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